Experimental Mathematics | 2021
An Interpolation From Sol to Hyperbolic Space
Abstract
We study a one-parameter family of nonisomorphic solvable Lie groups, which, when equipped with their left-invariant metrics, becomes an interpolation from a model of the Sol geometry to a model of Hyperbolic Space, with a stop at $\\mathbb{H}^2\\times\\mathbb{R}$. These Lie groups are also Bianchi groups of Type VI with orthogonal coordinates. As a continuation of joint work with Richard Schwartz on Sol, we primarily analyze those Lie groups in our interpolation with some positive sectional curvature.